[br]Below is a circle with an angle, [math]\theta[/math], and a radius, r. Move the point (r, [math]\theta[/math]) around and see what shape it creates. [br][br]Think about how x and y relate to r and [math]\theta[/math].
MIND CHECK: [br]Do you remember your trig and right triangle rules? Check them out here: [br][br]SOH CAH TOA[br][br]sin[math]\theta[/math] = [math]\frac{o}{h}[/math] = [math]\frac{y}{r}[/math][br]cos[math]\theta[/math] = [math]\frac{a}{h}[/math] = [math]\frac{x}{r}[/math][br][br]From these two things, with some moving around, we see that x = rcos[math]\theta[/math] and y = rsin[math]\theta[/math]. These equations help up convert polar coordinates, (r, [math]\theta[/math]) to cartesian coordinates (x, y).
From the above activity, we see that moving around the point (r, [math]\theta[/math]) gives us a circle if we go around [br]2[math]\pi[/math] radians, a full revolution. [br][br]With our conversion above, our circle equation, [math]x^2+y^2=r^2[/math] [math]\Longleftrightarrow[/math] [math]\left(rcos\theta\right)^2+\left(rsin\theta\right)^2=r^2cos^2\theta+r^2sin^2\theta=r^2\left(cos^2\theta+sin^2\theta\right)=r^2\left(1\right)=r^2[/math] and r = [math]\sqrt{x^2+y^2}[/math]. [br][br]Now we have Cartesian to Polar coordinate conversion equations.
Think about the equation r = a. What is this telling us about the circle it represents? Use the graph below to help you.
This equation is saying that no matter what angle we’ve got, the distance from the origin must be [i]a[/i]. If you think about it that is exactly the definition of a circle of radius [i]a [/i]centered at the origin.[br][br]So, this is a circle of radius [i]a[/i] centered at the origin. This is also one of the reasons why we might want to work in polar coordinates. The equation of a circle centered at the origin has a very nice equation, unlike the corresponding equation in Cartesian coordinates.[br][br]If r = [math]\sqrt{x^2+y^2}[/math] then [math]\sqrt{x^2+y^2}[/math] = a and so[math]x^2+y^2=a^2[/math], hence r = a is a circle centered at the origin with radius a. [br][math]\sqrt{x^2+y^2}[/math]
If we think about r = 2acos[math]\theta[/math] what is this telling us? Think about what x is in polar coordinates.[br][br]Play around with the circle below to figure out what this is telling us.
If you said this means that we have a circle with radius |a| centered at (a, 0) then you are thinking correctly.
What about r = 2bsin[math]\theta[/math]? Explore below:
If your exploration got you to see that this equation gives you a circle with radius |b| centered at (0, b) then you are seeing things correctly. [br][br]
Last but not least, let's think about r = 2acosθ + 2bsinθ[br]
This is the general equation of a circle!